The use of numerical numbers to represent molecular networks plays a crucial role in the study of physicochemical and structural properties of the chemical compounds. For some integer
k
and a network
G
, the networks
S
k
G
and
R
k
G
are its derived networks called as generalized subdivided and generalized semitotal point networks, where
S
k
and
R
k
are generalized subdivision and generalized semitotal point operations, respectively. Moreover, for two connected networks,
G
1
and
G
2
,
G
1
G
2
S
k
and
G
1
G
2
R
k
are
T
-sum networks which are obtained by the lexicographic product of
T
G
1
and
G
2
, respectively, where
T
ε
S
k
,
R
k
. In this paper, for the integral value
k
≥
1
, we find exact values of the first and second Zagreb indices for generalized
T
-sum networks. Furthermore, the obtained findings are general extensions of some known results for only
k
=
1
. At the end, a comparison among the different generalized
T
-sum networks with respect to first and second Zagreb indices is also included.